reserve R for Ring,
  r for Element of R,
  a, b, d1, d2 for Data-Location of R,
  il, i1, i2 for Nat,
  I for Instruction of SCM R,
  s,s1, s2 for State of SCM R,
  T for InsType of the InstructionsF of SCM R,
  k for Nat;

theorem Th35:
for k being Nat holds k+1 in SUCC(k,SCM R) &
 for j being Nat st j in SUCC(k,SCM R) holds k <= j
proof
  let k be Nat;
A1: SUCC(k,SCM R) = {k, k + 1} by Th34;
  hence k+1 in SUCC(k,SCM R) by TARSKI:def 2;
  let j be Nat;
  assume
A2: j in SUCC(k,SCM R);
  per cases by A1,A2,TARSKI:def 2;
  suppose
    j = k;
    hence thesis;
  end;
  suppose
    j = k + 1;
    hence thesis by NAT_1:11;
  end;
end;
